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Meantone family: Difference between revisions

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Comma: 81/80
Comma list: 81/80


[[Mappings|Period-generator mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]
[[Mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]


Mapping generator: ~3
Mapping generators: ~2, ~3


[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
[[Wedgie]]: {{wedgie| 1 4 4 }}


[[Tuning Ranges of Regular Temperaments|nice range]]: [694.786, 701.955] (1/3 comma to Pythagorean)
[[Tuning ranges]]:  


[[Tuning Ranges of Regular Temperaments|strict range]]: [694.786, 701.955]
* valid range: [685.714, 720.000] (7 to 5)
* nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
* strict range: [694.786, 701.955]


{{EDOs|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
{{EDOs|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
[[Wedgie]]: {{wedgie| 1 4 4 }}


[[Badness]]: 0.00736
[[Badness]]: 0.00736
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<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>


* [[Wikipedia: Septimal meantone temperament]]
{{main| Meantone }}
{{see also| Wikipedia: Septimal meantone temperament }}


The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone also tempers out the common 7-limit comma [[225/224]] and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.
The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone also tempers out the common 7-limit comma [[225/224]] and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.
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[[Comma]]s: 81/80, 126/125
[[Comma]] list: 81/80, 126/125


[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]


Mapping generator: ~3
Mapping generators: ~2, ~3


7 and [[9-odd-limit|9-limit]] minimax
[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}
 
[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]


[[Eigenmonzo]]s: 2, 5
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 2, 5


[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
[[Tuning ranges]]:
 
* valid range: [694.737, 700.000] (19 to 12)
nice range: [694.786, 701.955]
* nice range: [694.786, 701.955]
 
* strict range: [694.786, 700.000]
strict range: [694.786, 700.000]


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.


[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}
{{EDOs|legend=1| 12, 19, 31, 81, 112b, 143b }}
 
[[Vals]]: 12, 19, 31, 81, 112b, 143b


[[Badness]]: 0.0137
[[Badness]]: 0.0137
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[[Comma]]s: 81/80, 525/512
[[Comma]] list: 81/80, 525/512


[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]


Mapping generator: ~3
Mapping generators: ~2, ~3


[[7-odd-limit|7-limit]] minimax
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}


[{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }},
[[Minimax tuning]]:
{{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 7/5
* [[7-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
: [[Eigenmonzo]]s: 2, 7/5


[[9-odd-limit|9-limit]] minimax
* [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: Eigenmonzos: 2, 9/7


[{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }},
[[Tuning ranges]]:  
{{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}<nowiki>]</nowiki>
* valid range: [692.308, 694.737] (26 to 19)
 
* nice range: [692.353, 701.955]
Eigenmonzos: 2, 9/7
* strict range: [692.353, 694.737]
 
valid range: [692.308, 694.737] (26 to 19)
 
nice range: [692.353, 701.955]
 
strict range: [692.353, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}


{{EDOs|legend=1| 7, 19, 26, 45 }}
{{EDOs|legend=1| 7, 19, 26, 45 }}
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[[Comma]]s: 49/48, 81/80
[[Comma]] list: 49/48, 81/80
 
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 2}}, {{val|0 2 8 1}}]
 
Mapping generator: ~7/4


valid range: [240.000, 257.143] (5 to 14c)
[[Mapping]]: [{{val| 1 0 -4 2 }}, {{val| 0 2 8 1 }}]


nice range: [231.174, 266.871]
Mapping generators: ~2, ~7/4


strict range: [240.000, 257.143]
[[Wedgie]]: {{wedgie| 2 8 1 8 -4 -20 }}


[[Wedgie]]: &lt;&lt;2 8 1 8 -4 -20||
[[Tuning ranges]]:  
* valid range: [240.000, 257.143] (5 to 14c)
* nice range: [231.174, 266.871]
* strict range: [240.000, 257.143]


{{EDOs|legend=1| 5, 14c, 19 }}
{{EDOs|legend=1| 5, 14c, 19 }}
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Commas: 81/80, 121/120, 176/175
Comma list: 81/80, 121/120, 176/175


Period-generator mapping: [{{val|1 1 0 6 2}}, {{val|0 2 8 -11 5}}]
Mapping: [{{val|1 1 0 6 2}}, {{val|0 2 8 -11 5}}]


Mapping generator: ~11/9
Mapping generators: ~2, ~11/9
 
[[11-odd-limit|11-limit]] minimax 1/4 comma
 
[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }},
{{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}<nowiki>]</nowiki>


Eigenmonzos: 2, 5
Minimax tuning:
* [[11-odd-limit]]: 1/4 comma
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: Eigenmonzos: 2, 5


{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
{{EDOs|legend=1| 7, 24, 31, 38, 55 }}
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